Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1555 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(6,10)(7,14)(8,16)(11,13)(12,18)(15,20);(5,6,7,12,16,13,20,19,9,11,15,18,8,10,14,17);(1,3,4,2)(6,11)(7,15)(10,13)(14,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=ghi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#128918, #72217755, #72217768, … |
| maximal normal subgroup of | C2×#128918, #72217755, #72217768, … |
| maximal quotient of | #72216848, C2×#128918, #72221585 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36114);